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Impossible world/Hintikka: I believe that we must allow the impossible world to fight the problem of another kind of omniscience, the logical omniscience.
Impossible worlds/Logical omniscience/semantics of possible worlds/Hintikka: Thesis: the problem of omniscience does not occur here at all! E.g.
(1) A sentence of the form "a knows that p" is true in a world W iff. P is true in all a-alternatives. That is, in all worlds, which are compatible with the knowledge of a.
Logical omniscience: their failure can be formulated like this:
(2) There is a, p and q such that a knows that p, p implies logically q, but a does not know that q.
Logical truth: is then analyzed model-theoretically:
(3) A sentence is logically true, iff. it is true in every logically possible world.
Problem: (1) - (3) are incompatible! However, they are not yet incompatible in the form given above, but only with the additional assumption:
(4) Every epistemically possible world is logically possible.
Problem: now it can be that in an epistemic a-alternative W'q is wrong!
Problem: According to (4), these epistemic worlds are also logically possible.
However, according to the logical truth of (p > q) ((s) in this example), q must be true in any logically possible world. This results in the contradiction.
Solution: different authors have responded differently:
Positivism: positivism takes refuge in the noninformative (tautological) logical truth.
HintikkaVs: instead: semantics of possible worlds.
(4): already presupposes omniscience! It assumes that a can only eliminate seeming possibilities. This is circular.
Solution: there may be possibilities that appear only possible but contain hidden contradictions.
Problem: the problem here is (4) and not (2)!
Solution/Hintikka: we have to allow worlds that are logically impossible, but still epistemically possible. ((s) unlike the impossible worlds discussed in Stalnaker and Cresswell.)
Then (1) - (3) can be true together. That is, in an epistemic world (p > q) can fail.
Impossible world/Hintikka: Problem as how we can allow it.
Impossible world/Cresswell/Hintikka: Cresswell proposes a reinterpretation of the logical constants. (Model theoretical).
HintikkaVsCresswell: the real problem with omniscience is that people do not recognize all the logical consequences of their knowledge. And this takes place in classical logic. Non-standard logic: bypasses the problem. You could say it destroys the problem instead of solving it.
Impossible world/Logical omniscience/Solution/Veikko RantalaVsHintikka: has solved some problems of this approach.
Nonclassical models: nonclassical models are for first level sentences.
Impossible world/Rantala: are not "impossible" according to Rantala, but they differ from normal possible worlds, in the way that they are "changing worlds" by allowing new individuals.
However, in such a subtle way that they normally cannot be distinguished from invariant worlds (with always the same individuals). It is about:
Urn model/Statistics/Omniscience/Hintikka: whereby the variant worlds are such worlds with which moves from the urn possibly get new individuals into the game. But only so few that you may not notice it.
Jaakko and Merrill B. Hintikka
The Logic of Epistemology and the Epistemology of Logic Dordrecht 1989
J. Hintikka/M. B. Hintikka
Untersuchungen zu Wittgenstein Frankfurt 1996